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New perspectives on polygonal and polyhedral finite element methods (2014)
| Venue: | MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES |
| Citations: | 5 - 2 self |
BibTeX
@MISC{Manzini14newperspectives,
author = {Gianmarco Manzini and Alessandro Russo and N. Sukumar},
title = {New perspectives on polygonal and polyhedral finite element methods},
year = {2014}
}
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Abstract
Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finite-difference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the Virtual Element Method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more in-depth understand-ing of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semi-definite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and poly-hedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates post-processing of field variables and
Keyphrases
polyhedral finite element method mathematical model consistent matrix applied science generalized barycentric coordinate stability matrix previous development virtual element method polygonal-based galerkin method exact decomposition poly-hedral finite element arbitrary polygonal mesh in-depth understand-ing mimetic finite-difference scheme stiffness matrix explicit construction mean value coordinate barycentric coordinate-based galerkin method four-node finite element method field variable variational framework polyhedral element numerical quadrature positive semi-definite mimetic scheme stable finite element method basis function projection operator
